Lecture Schedule: Difference between revisions
Created page with "====== Lectures ======= Tentative lecture plan is as follows: === Week 1 === Topics: Review of differential geometry * Topological and smooth manifolds, * Vector and tensor fields, * Algebra of differential forms, * De Rham Cohomology Reading: * Taubes Ch 1 * Frenkel Sections 1-3 * Nakahara Sections 5.1, 5.2, 5.4. 6. * Sontz Chapters 2, 3, 5 === Week 2 === Topics: Review of differential geometry * vector fields * Lie brackets * diffeomorphism..." |
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== Lectures == | |||
Tentative lecture plan is as follows: | Tentative lecture plan is as follows: | ||
=== Week 1 === | === Week 1 === | ||
Topics: Review of differential geometry | Topics: Review of differential geometry | ||
* [[Topological and smooth manifolds]], | |||
* Vector and tensor fields, | |||
* Algebra of differential forms, | |||
* De Rham Cohomology | |||
Reading: | Reading: | ||
* Taubes Ch 1 | |||
* Frenkel Sections 1-3 | |||
* Nakahara Sections 5.1, 5.2, 5.4. 6. | |||
* Sontz Chapters 2, 3, 5 | |||
=== Week 2 === | === Week 2 === | ||
Topics: Review of differential geometry | Topics: Review of differential geometry | ||
* vector fields | |||
* Lie brackets | |||
* diffeomorphisms | |||
* Lie derivatives | |||
* Lie groups and Lie algebras. | |||
* Exponential maps | |||
* adjoint representation | |||
* Basics of Symplectic Geometry | |||
* Hamiltonian vector fields | |||
* Hamiltonian group actions | |||
* Moment maps and co-moment maps. | |||
Reading: | Reading: | ||
* Rudolph-Schmidt (Differential Geometry and Mathematical Physics. Part I) Section 5 | |||
=== Week 3 === | === Week 3 === | ||
Topics: | Topics: | ||
* Further details on smooth actions, | |||
* Symplectic structure on cotangent bundles | |||
* Quotients of free, proper, smooth actions | |||
* Principal bundles. | |||
* Frame bundles. | |||
* Associated fibre bundles. | |||
| Line 47: | Line 51: | ||
=== Week 4 === | === Week 4 === | ||
Topics: | Topics: | ||
* Associated fibre bundles: vector bundles, adjoint bundle. | |||
* Reduction and extension of structure group. | |||
* Group of gauge transformations. | |||
* Linear connections. | |||
Reading: | Reading: | ||
=== Week 5 | === Week 5=== | ||
Topics: | Topics: | ||
* Connections: Koszul connections, principal connections, Ehresmann connections. | |||
* Equivalences between them. | |||
* Atiyah sequence and differential operators. | |||
Reading: | Reading: | ||
Latest revision as of 08:54, 19 January 2026
Lectures
[edit]Tentative lecture plan is as follows:
Week 1
[edit]Topics: Review of differential geometry
- Topological and smooth manifolds,
- Vector and tensor fields,
- Algebra of differential forms,
- De Rham Cohomology
Reading:
- Taubes Ch 1
- Frenkel Sections 1-3
- Nakahara Sections 5.1, 5.2, 5.4. 6.
- Sontz Chapters 2, 3, 5
Week 2
[edit]Topics: Review of differential geometry
- vector fields
- Lie brackets
- diffeomorphisms
- Lie derivatives
- Lie groups and Lie algebras.
- Exponential maps
- adjoint representation
- Basics of Symplectic Geometry
- Hamiltonian vector fields
- Hamiltonian group actions
- Moment maps and co-moment maps.
Reading:
- Rudolph-Schmidt (Differential Geometry and Mathematical Physics. Part I) Section 5
Week 3
[edit]Topics:
- Further details on smooth actions,
- Symplectic structure on cotangent bundles
- Quotients of free, proper, smooth actions
- Principal bundles.
- Frame bundles.
- Associated fibre bundles.
Reading:
Week 4
[edit]Topics:
- Associated fibre bundles: vector bundles, adjoint bundle.
- Reduction and extension of structure group.
- Group of gauge transformations.
- Linear connections.
Reading:
Week 5
[edit]Topics:
- Connections: Koszul connections, principal connections, Ehresmann connections.
- Equivalences between them.
- Atiyah sequence and differential operators.
Reading: